Before watching the video, the substitution I made was pi/2*x = u, then used the double angle formula for sin in the denominator of the integrand. In the numerator, I used the fact that sin^2 u + cos^2 u = 1, then integrated by parts.

@monikaherath7505

11 ай бұрын

Could you write it out please I don't understand how you got to sin^2 u + cos*2 u

@The1RandomFool

11 ай бұрын

sin^2 u + cos^2 u = 1 is a basic trigonometric identity. In general, it's a strategy to cancel out a product of sines and cosines in the denominator by setting 1 = sin^2 u + cos^2 u in the numerator.@@monikaherath7505

Using Euler reflexion formula you can also show that 7ζ(3)/pi^2 = integral(gamma(3/2-x)*gamma(3/2+x)) from -1/2 to 1/2. It doesn't help to integrate, but it's beautiful ;-)

GREAT!! An interesting class of integrals: integrate from 0 to 1 [ x^n (1-x)^m / sin(πx)].

Thank you for this innovative integration. Smart solution.

Awesome integral!

Bro was like LOL do it by parts and stuff but really didn't bother by going the distance, great video as always brother.

@maths_505

11 ай бұрын

Thanks mate. I've been looking for an integral to use that series on

Hi, use the properties of the gamma function like is relation to 1/sin(x.pi) and x(x-1)

Bring back dark thumbnail 1/4

@maths_505

11 ай бұрын

Okay tomorrow

Nice one ! I now challenge you to solve the integral from 0 to pi of x * (sinx)^n dx. It took me around an hour or 2 to solve and is absoluetly gorgeous

@grigoriefimovitchrasputin5442

11 ай бұрын

Thanks ! It looks interesting to solve

@quentinrenon9876

11 ай бұрын

@@grigoriefimovitchrasputin5442 Tbh it's not too difficult but you need to be creative, normal methods don't work. Or at least I couldn't get very far using them

@grigoriefimovitchrasputin5442

11 ай бұрын

@@quentinrenon9876 it looks like Wallis integrals, except that there is a x. Anyway, i'll give it a try

@alarka1782

11 ай бұрын

I think by applying property of integral, it breaks into pi * integral 0 to pi/2 (sin x) ^ n dx. Should be easy from there to express in terms of gamma function.

@user-uh9bo2im1h

11 ай бұрын

Should be doable by integrating by x = sqrt(x) then u = x/2 and then letting u be sin x which leads to a structure very similar to the beta function can’t rlly go further because I have no pen rn

Isn't the essence of the proof the following Fourier Expansion for csc(x) = 2 SUM_odd (sin(nx))?

It looks like it can be generalised. If n=3, then ans can be written as (2n+1)zeta(n)/pi^{n}. May be exist some types of integrals which are generalised that way?

@austin4768

4 ай бұрын

Yes I’m also curious if there are analogous formulas for zeta of odd (or even all) n. One can go try to go back through the calculation and find a point where the integrand can be tweaked in such a way to yield this kind of formula. I’m too lazy to do this right now though

Corretto, ❤ho usato sinx, formula esponenziale, poi la serie geometrica... Anche se a me risulta un segno -... Ah ah, al primo colpo non mi viene mai... Ok, il segno - è scomparso... Is correct

cr7 fan here too xD, loved that one 👍

@maths_505

11 ай бұрын

SUIIIIIIIIIIIIIIIIIIIII

Didn't go the way I thought

I love your solution though I don't understand one of your solution. I'm an A level student.😊

@maths_505

11 ай бұрын

Oh cool....I used to tutor A level students..... Thanks bro....you'll understand them all pretty soon cuz you'll study the basics in your A levels....then you'll solve these integrals and DEs on your own.

How is this applied to every day life?

@insouciantFox

11 ай бұрын

Integrals are everyday life

@daddy_myers

11 ай бұрын

Integrals are life.

@damrgee8279

11 ай бұрын

@@daddy_myers example please

@daddy_myers

11 ай бұрын

@@damrgee8279 Your question alone indicates you could use more integrals in your daily life.

@damrgee8279

11 ай бұрын

@Jacques-kc4qy it’s amazing that people like yourself are incredibly smart when it comes to this stuff I don’t understand any of it, But yet when it comes to political leanings regarding logic and common sense we have some of the stupidest people on the planet